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# Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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### Prove that the function $F(f)=\int_a^bf(x)dx$ is continuous. [closed]

Define $d:X\times X\to \mathbb{R}$ by $$d(f,g)=\int_a^b |f(x)-g(x)|dx$$ Let $\rho$ be the usual metric on $\mathbb{R}$. Prove that the function $F:X\to\mathbb{R}$ defined $$F(f)=\int_a^bf(x)dx$$ is ...
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### Example of a locally-convex topological vector space which is not metrizable

I seek an example of a locally-convex topological vector space which is not a metric space. From google I found an example LF-Space. Does there exist other examples ?
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### Let $(A_{j})$ be a sequence of closed subsets of $X$ with $A_{j} \supseteq A_{j+1}$ for all $j \in \mathbb{N}$, then $\cap A_{j} \neq \emptyset .$

Good afternoon, I'm doing Problem III.3.5 from textbook Analysis I by Amann. Here the underlying space is metric space. Could you please verify if my proof looks fine or contains logical ...
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### In an ultrametric space, is every open set closed?

I saw the following well-known fact for ultrametric spaces Every open ball is closed. So this stimulates me to think whether this is true for open set or not. By an ultramtric space, it's a ...
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### Singular matrices are Nowhere Dense

I need to show that the subset S of all singular matrices is Nowhere dense in Set of all matrices of order n. What I tried:- As I need to show closure of S has no interior point . But S is a closed ...
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### Proving function is well-defined over set of equivalence classes

This is a subproblem of a problem I am trying to solve. For a measure space $(X,\mathcal{A}, \mu)$ define $\backsim$ as the relation on $\mathcal{A}$ where $A \backsim B$ iff $\mu(A \triangle B)=0$ ...
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### $d(A,B)=\min\{ d(a,b): a\in A\text{ and }b\in B\}>0$ then $A\cup B$ is not connected

If $d(A,B)=\min\{ d(a,b): a\in A\text{ and }b\in B\}>0$ then $A\cup B$ is not connected. We claim that the two sets are disjoint, suppose it is not we have $d(A,B)=0$ So now we have to say any ...
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### Subspace of a separable metric space is also separable [duplicate]

I see a result on a textbook but I do not know how to prove it. let $(X,d)$ be a separable metric space and $A \subset X$. Suppose that the topology induced on $A$ by the metric $d_A$ coincides with ...
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### Convergence of sequences in metric space

For each positive integer $m$, consider the following sequences: $x_m:=\{0,0,...0,m,m,m,...\}$ and $y_m:=\{0,...,0,\frac{1}{m}, \frac{1}{m},...\}$, where the only zero terms of these sequences are ...
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### If $A_k \subseteq X_k$ is closed in $X_k$ for all $1 \leq k \leq m$, then $\prod_{k=1}^{m} A_k$ is closed in $\prod_{k=1}^{m} X_k$

I'm trying to prove this basic property of the product metric. Could you please verify if my proof looks fine or contains logical gaps/errors? Thank you so much for your help! If $(X_{k}, d_{k})$ are ...
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### The canonical projections on metric spaces are continuous and open

Bonjour, I'm doing Problem III.2.18 from textbook Analysis I by Amann/Escher. $p$ is open if the images of open sets under $p$ are open. $X \times Y$ is endowed with the product metric. ...
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### Show that $M(3)$ of metric axioms can be replaced by a weaker axiom $M'(3)$

$M(3):=\text{ For all$x,y,z \in X$,$d(x,z) \leq d(x,y)+d(y,z)$}$ $M'(3):= \text { If$x,y,z \in X$are distinct, then$d(x,z)\leq d(x,y)+d(y,z)$}$. [all the other axioms are identical] Proof: To ...
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